(Grothendieck) topoi are left-exact reflective subcategories of a category of presheaves. An important class of quasi-topoi (see: http://ncatlab.org/nlab/show/quasitopos) arise as the category of concrete sheaves on a concrete site. Concrete sheaves are those sheaves $X$ such that the induced map $Hom(C,X) \to Hom(\underline{C},\underline{X})$ is injective for all objects $C$, where $\underline{C}$ is the underlying set of $C$ and $\underline{X}$ is the value of $X$ on the terminal object. Concrete sheaves are a reflective subcategory of all sheaves. Concrete sheaves are a particularly nice example of a quasi-topos as the resulting quasi-topos is both complete and cocomplete. My question is, is there a way to represent quasi-topoi (or nice ones) as reflective subcategories of a Grothendieck topos (with some condition on the reflector)? (Of course, for this, you'd need the quasi-topos to be complete, since reflective subcategories of complete categories are again complete). More generally, is there some theorem saying that a category is a (possibly non-complete) quasi-topos if and only if it can be embedded into a topos such that the embedding has such-and-such property?
[Math] Classification of Quasi-topoi
ct.category-theorytopos-theory
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The profinite fundamental group of $X_{fppf}$ as you define it is again the etale fundamental group of X. More precisely, the functor (of points)
$f : X_{et} \to \mathrm{Sh}_{fppf}(X)$
is fully faithful and has essential image the locally finite constant sheaves (image clearly contained there, as finite etale maps are even etale locally finite constant, let alone fppf locally so). Proof in 3 steps:
It is fully faithful by Yoneda (note also well-defined by fppf descent for morphsisms).
Both sides are fppf sheaves (stacks) in $X$, by classical fppf descent.
Combining 1 and 2, it suffices to show that a sheaf we want to hit is just fppf locally hit, which is obvious since locally it's finite constant.
Note that the same proof also works for $X_{et}$ or anything in between -- once your topology splits finite etale maps it doesn't really matter what it is. So we usually just work with the minimal one, the small etale topology. As Mike Artin said to me apropos of something like this, "Why pack a suitcase when you're just going around the corner?"
I want to give two simple observations about diffeological spaces that might provide a partial answer to your question.
1) We have the following inclusions of full subcategories $$Mfd \subset Diff \subset Sh \subset PSh$$ where $Mfd$ is the category of smooth finite dim manifolds, $Diff$ are diffeological spaces (i.e. concrete sheaves on cartesian spaces), Sh are sheaves on cartesian spaces and $PSh$ are presheaves on cartesian spaces. The last two inclusions are reflexive.
Lets us first have a closer look at the inclusion $Sh \subset PSh$. Following the same vein of argument as above, there is a priori no reason to work with $Sh$ instead of $PSh$ since both categories are equally nice (topoi) and the definition of a presheaf is clearly simpler than that of a sheaf. But there are some colimits in $Mfd$ that we really like, namely the coequalizer diagram correspoding to an open cover $(U_\alpha)$ of a manifold $M$. Under the inclusion of $Mfd$ into $PSh$ this is not a coequalizer anymore, in other words: If we glue open sets in $PSh$ together we do not get the same thing that we get when glueing together as manifolds. This defect is exactly cured by the sheaf property. That means restricting to the smaller subcategory $Sh \subset PSh$ the colimits change such that gluing of open sets behaves as nice as in manifolds. The punchline is that the restriction to $Sh$ provides the category with the "right" coequalizers of open sets.
Now lets turn towards the inclusion $Diff \subset Sh$. The situation is exactly the same as before. Limits in $Diff$ are computed as Limits in $Sh$ (and hence also $PSh$) but colimits are different in general (one has to apply the concretization functor). This is what happens categorically. Now it turns out that there are colimits in manifolds that become colimits in diffeological spaces but not colimits in sheaves. Here an example would be very nice. Unfortunately I have not been able the remember the example I had for this behaviour. Even so, from abstract reasoning it is clear that the colimits in the two categories have to differ.
Hence one could argue that diffeological spaces have the right "geometric" colimits and sheaves do not. The price is of course that we exclude some interesing "spaces" like the sheaf of diffential forms and loose the property that the category is a topos.
2) If we want to "make" geometry over diffeological spaces it turns out that there are two possible definition of principal bundles:
a bundle over a diffeological space $M$ is a morphism to the stack of bundles over finite dimensional manifolds. This means that we have a family of bundles over each plot together with coherent isomorphisms. Note that this type of bundle is determined by its pullback to finite dimensional spaces. This is equivalent to have a diffeological space $P \to M$ together with a free transitive on fibers action such that the quotient map $P \to M$ is a surjective subduction (i.e. becomes a submersion on each plot). To get those type of bundles we have to equip diffeological spaces with the Grothendieck Topology of subductions.
a bundle over a diffeological space $M$ is a space $P \to M$ with a free, transitive on fibers, action such that it is locally trivial, where locally refers to the underlying topological space of $M$. This is the type of bundle which people consider in the world of $\infty$-dimensional manifolds. To get this we have to take the grothendieck topology of morphisms that are surjective and admits local (in the topology) sections. Hence therefore we really need the underlying topological space.
I do not prefer one of the two possible Grothendieck Topologies, but the second one is closer to what people have done in the $\infty$-dimensional setting. And one can show that the universal bundle $EG \to BG$ for a compact Lie-group is of this type (of course one has to find diffeological models of $BG$ and $EG$).
The first topology has an obvious analogue on the category $Sh$ of all sheaves but the second crucially uses the underlying topological space of a diffeological space.
Best Answer
This is only a partial answer, and you may know it already since I mentioned it recently at the nForum, but for completeness, here it is. Theorem C2.2.13 in Sketches of an Elephant shows that the following are equivalent for a category C:
A category of this sort is called a Grothendieck quasitopos. The third characterization seems most similar to what you're looking for. I doubt you can get away without some generating-set condition, since it seems very unlikely that the (complete, cocomplete, locally small) quasitopos of pseudotopological spaces (for example) can be reflectively embedded in a topos.
What I don't know is whether one can put conditions directly on a reflective subcategory of a topos, analogous to left-exactness of the reflector, to guarantee that it is of this form. The reflector for separated objects preserves finite products and monics, but I have no idea whether that would be sufficient as a characterization.